Symmetries, Conservation Laws, and Exact Solutions of a Potential Kadomtsev–Petviashvili Equation with Power-Law Nonlinearity
Abstract
1. Introduction
2. Conservation Laws Associated with Equation (2)
- Case 1: p is arbitrary but
- Case 2:
- Case 3:
- Case 4:
3. Symmetry Reduction of Equation (2)
3.1. Case
3.2. Case
4. Exact Solution of (2) Using the Kudryashov Method and Direct Integration
- Step 1. The transformation , where k and are constants, reduces Equation (64) to the ordinary differential equation (ODE)
- Step 2. It is assumed that the exact solution of Equation (65) can be expressed by a polynomial in Q as follows:We note that Equation (67) has the solution given byThe positive integer N is determined by taking the pole order of general solution for Equation (65). Substituting into monomials of Equation (65) and comparing the two or more terms with smallest powers in the equation, we find the value for
- Step 3. We substitute the derivatives of with respect to z and the expression for into Equation (65), and as a result, we obtain the equation that has the function Q, coefficients , and parameters of Equation (65).
- Step 4. The KM effectively reduces the task of finding an exact solution to the ordinary differential equation (ODE) (65) to solve a corresponding system of algebraic equations. By substituting the assumed solution form into the ODE and collecting like powers of the function one obtains a polynomial expression in Setting the coefficients of each power of Q to zero yields a system of algebraic equations, which can then be solved to determine the parameters of the exact solution. The resulting system takes the form
- Step 5. Solving the system of algebraic equations, we obtain values of coefficients and relations for the parameters of Equation (65). As a result of the solution, we obtain exact solutions of Equation (65) in the form (66).
- Step 6. We present the solution of Equation (65) in a more convenient form and verify the solutions.
4.1. Solutions of (2) with Using the Kudryashov Method
4.2. Solutions of (2) with Using Direct Integration
5. Conclusions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
Abbreviations
NLPDE | Nonlinear Partial Differential Equation |
NLPDEs | Nonlinear Partial Differential Equations |
PDEs | Partial Differential Equations |
PDE | Partial Differential Equation |
ODEs | Ordinary Differential Equations |
ODE | Ordinary Differential Equation |
KDV | Korteweg–de Vries |
KP | Kadomtsev–Petviashvili |
PKP | Potential Kadomtsev–Petviashvili |
PKPp | Potential Kadomtsev–Petviashvili with Power-Law Nonlinearity |
VCPKP | Variable Coefficient Potential Kadomtsev–Petviashvili |
KM | Kudryashov Method |
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Mothibi, D.M. Symmetries, Conservation Laws, and Exact Solutions of a Potential Kadomtsev–Petviashvili Equation with Power-Law Nonlinearity. Symmetry 2025, 17, 1053. https://doi.org/10.3390/sym17071053
Mothibi DM. Symmetries, Conservation Laws, and Exact Solutions of a Potential Kadomtsev–Petviashvili Equation with Power-Law Nonlinearity. Symmetry. 2025; 17(7):1053. https://doi.org/10.3390/sym17071053
Chicago/Turabian StyleMothibi, Dimpho Millicent. 2025. "Symmetries, Conservation Laws, and Exact Solutions of a Potential Kadomtsev–Petviashvili Equation with Power-Law Nonlinearity" Symmetry 17, no. 7: 1053. https://doi.org/10.3390/sym17071053
APA StyleMothibi, D. M. (2025). Symmetries, Conservation Laws, and Exact Solutions of a Potential Kadomtsev–Petviashvili Equation with Power-Law Nonlinearity. Symmetry, 17(7), 1053. https://doi.org/10.3390/sym17071053